### Abstract

A moduli space of sheaves satisfies weak Brill-Noether if the general sheaf in the moduli space has no cohomology. Göttsche and Hirschowitz prove that on P^{2} every moduli space of Gieseker semistable sheaves of rank at least two and Euler characteristic zero satisfies weak Brill-Noether. In this paper, we give sufficient conditions for weak Brill-Noether to hold on rational surfaces. We completely characterize Chern characters on Hirzebruch surfaces for which weak Brill-Noether holds. We also prove that on a del Pezzo surface of degree at least 4 weak Brill-Noether holds if the first Chern class is nef.

Original language | English (US) |
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Title of host publication | Contemporary Mathematics |

Publisher | American Mathematical Society |

Pages | 81-104 |

Number of pages | 24 |

DOIs | |

State | Published - Jan 1 2018 |

### Publication series

Name | Contemporary Mathematics |
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Volume | 712 |

ISSN (Print) | 0271-4132 |

ISSN (Electronic) | 1098-3627 |

### All Science Journal Classification (ASJC) codes

- Mathematics(all)

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## Cite this

Coskun, I., & Huizenga, J. (2018). Weak brill-noether for rational surfaces. In

*Contemporary Mathematics*(pp. 81-104). (Contemporary Mathematics; Vol. 712). American Mathematical Society. https://doi.org/10.1090/conm/712/14343